Download pdf - A weak Galerkin mixed finite element method for second ... · A WEAK GALERKIN MIXED FINITE ELEMENT METHOD FOR SECOND ORDER ELLIPTIC PROBLEMS JUNPINGWANGANDXIUYE Abstract. A new weak

MATHEMATICS OF COMPUTATIONVolume 83, Number 289, September 2014, Pages 2101–2126S 0025-5718(2014)02852-4Article electronically published on May 5, 2014

A WEAK GALERKIN MIXED FINITE ELEMENT METHOD

FOR SECOND ORDER ELLIPTIC PROBLEMS

JUNPING WANG AND XIU YE

Abstract. A new weak Galerkin (WG) method is introduced and analyzedfor the second order elliptic equation formulated as a system of two first orderlinear equations. This method, called WG-MFEM, is designed by using dis-continuous piecewise polynomials on finite element partitions with arbitraryshape of polygons/polyhedra. The WG-MFEM is capable of providing veryaccurate numerical approximations for both the primary and flux variables.

Allowing the use of discontinuous approximating functions on arbitrary shapeof polygons/polyhedra makes the method highly flexible in practical compu-tation. Optimal order error estimates in both discrete H1 and L2 norms areestablished for the corresponding weak Galerkin mixed finite element solutions.

1. Introduction

Weak Galerkin (WG) refers to a finite element technique for partial differentialequations in which differential operators are approximated by their weak forms asdistributions. In [29,30], a weak Galerkin method was introduced and analyzed forsecond order elliptic equations and the Stokes equation based on weak gradientsand weak divergence. In this paper, we shall develop a new weak Galerkin methodfor second order elliptic equations formulated as a system of two first order linearequations. Our model problem seeks a flux function q = q(x) and a scalar functionu = u(x) defined in an open bounded polygonal or polyhedral domain Ω ⊂ Rd (d =2, 3) satisfying

αq+∇u = 0, ∇ · q = f, in Ω(1.1)

and the following Dirichlet boundary condition

(1.2) u = −g on ∂Ω,

where α = (αij(x))d×d ∈ [L∞(Ω)]d2

is a symmetric, uniformly positive definitematrix on the domain Ω. A weak formulation for (1.1)-(1.2) seeks q ∈ H(div,Ω)

Received by the editor March 20, 2012 and, in revised form, November 23, 2012 and December11, 2012.

2010 Mathematics Subject Classification. Primary, 65N15, 65N30, 76D07; Secondary, 35B45,35J50.

Key words and phrases. Weak Galerkin, finite element methods, discrete weak divergence,second order elliptic problems, mixed finite element methods.

The research of the first author was supported by the NSF IR/D program, while working atthe Foundation. However, any opinion, finding, and conclusions or recommendations expressedin this material are those of the author and do not necessarily reflect the views of the NationalScience Foundation.

This research was supported in part by National Science Foundation Grant DMS-1115097.

c©2014 American Mathematical Society

2101

Licensed to Univ of Pittsburgh. Prepared on Fri Mar 4 12:28:22 EST 2016 for download from IP 136.142.124.50.

License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use

http://www.ams.org/mcom/

http://www.ams.org/jourcgi/jour-getitem?pii=S0025-5718-2014-02852-4

2102 JUNPING WANG AND XIU YE

and u ∈ L2(Ω) such that

(αq,v)− (∇ · v, u) = 〈g,v · n〉∂Ω, ∀v ∈ H(div,Ω),(1.3)

(∇ · q, w) = (f, w), ∀w ∈ L2(Ω).(1.4)

Here L2(Ω) is the standard Hilbert space of square integrable functions on Ω, ∇·vis the divergence of vector-valued functions v on Ω, H(div,Ω) is the Sobolev spaceconsisting of vector-valued functions v such that v ∈ [L2(Ω)]d and ∇ · v ∈ L2(Ω),(·, ·) stands for the L2-inner product in L2(Ω), and 〈·, ·〉∂Ω is the inner product inL2(∂Ω).

Galerkin methods based on the weak formulation (1.3)-(1.4) and finite dimen-sional subspaces of H(div,Ω) × L2(Ω) with piecewise polynomials are known asmixed finite element methods (MFEM). MFEMs for (1.1)-(1.2) treat q and u asunknown functions and are capable of providing accurate approximations for bothunknowns [1, 3, 7–11, 27, 28]. All the existing MFEMs in literature possess localmass conservation that makes MFEM a competitive numerical technique in manyapplications such as oil reservoir and groundwater flow simulation in porous me-dia. On the other hand, MFEMs are formulated in subspaces of H(div,Ω)×L2(Ω)which requires a certain continuity of the finite element functions for the flux vari-able. More precisely, the flux functions must be sufficiently continuous so that theusual divergence is well-defined in the classical sense in L2(Ω). This continuity as-sumption in turn imposes a strong restriction on the structure of the finite elementpartition and the piecewise polynomials defined on them [9–11,27, 30].

The weak Galerkin method introduced in [29] (see also [22, 23] for extensions)was based on the use of weak gradients in the following variational formulation:find u ∈ H1(Ω) such that u = −g on ∂Ω and

(1.5) (α−1∇u,∇φ) = (f, φ), ∀φ ∈ H10 (Ω),

whereH1(Ω) is the Sobolev space consisting of functions for which all partial deriva-tives up to order one are square integrable, H1

0 (Ω) is the subspace of H1(Ω) con-sisting of functions with vanishing value on ∂Ω. Specifically, the weak Galerkinfinite element formulation in [22, 23, 29, 30] can be obtained from (1.5) by simplyreplacing the gradient ∇ by a discrete gradient ∇w (it was denoted as ∇d in [29])defined by a distributional formula. The discrete weak gradient operator ∇w islocally-defined on each element. It has been demonstrated [24–26, 29, 30] that theweak Galerkin method enjoys an easy-to-implement formulation that is parameterfree and inherits the physical property of mass conservation locally on each element.Furthermore, the weak Galerkin method has the flexibility of using discontinuousfinite element functions, as was commonly employed in discontinuous Galerkin andhybridized discontinuous Galerkin methods [2, 15].

The goal of this paper is to extend the weak Galerkin method of [22, 23, 29] tothe variational formulation (1.3)-(1.4) by following the idea of weak gradients. Itis clear that divergence is the principle differential operator in (1.3)-(1.4). Thus,an essential part of the extension is the development of a weakly-defined discretedivergence operator, denoted by (∇w·), for a class of vector-valued weak functions ina finite element setting. Assuming that there is such a discrete divergence operator(∇w·) defined on a finite element space Vh for the flux variable q, then formallyone would have a WG method for (1.1)-(1.2) that seeks qh ∈ Vh and uh ∈ Wh



A WEAK GALERKIN MFEM FOR SECOND ORDER ELLIPTIC PROBLEMS 2103

satisfying

(αqh,v)− (∇w · v, uh) = 〈g,v · n〉∂Ω, ∀v ∈ Vh,

(∇w · qh, w) = (f, w), ∀w ∈ Wh,

where Wh is a properly defined finite element space for the scalar variable. Therest of the paper will provide details for a rigorous interpretation and justificationof the above formal WG method, which shall be called WG-MFEM methods.

The WG-MFEM have the following features. First of all, the finite elementpartition of the domain Ω is allowed to consist of arbitrary shape of polygonsfor d = 2 and polyhedra for d = 3. Second, the flux approximation space Vh

consists of two components, where the first one is given by piecewise polynomialson each polygon/polyhedra and the second is given by piecewise polynomials onthe edges/faces of the polygon/polyhedra. The second component shall be usedto approximate the normal component of the flux variable q on each edge/face.Furthermore, the scalar approximation space Wh consists of piecewise polynomialson each polygon/polyhedra with one degree higher than that of the flux. Forexample, the lowest order of such elements would consist of piecewise constant forflux and its normal component on each edge/face plus piecewise linear function forthe scalar variable on each polygon/polyhedra. There is no continuity required forany of the finite element functions in WG-MFEM.

One close relative of the WG-MFEM is the mimetic finite difference (MFD)method, see [4–6,12] and the references cited therein. Both WG-MFEM and MFDshare the same flexibility of using polygonal/polyhedral elements of the domain.The lowest order MFD approximates the flux by using only piecewise constantson each edge/face, and the scalar variable by using another piecewise constantfunction on each polygonal/polyhedral element, while WG-MFEM provides a wideclass of numerical schemes with arbitrary order of polynomials. The arbitrary-ordermimetic scheme of [4] is based on the philosophy of approximating the forms in(1.5) using nodal-based polynomials, which has very minimal overlap with the WG-MFEM. It should be pointed out that there are other numerical methods designedon general polygonal meshes in existing literature [2, 15, 17–20]. In particular, acomparison with existing numerical schemes should be conducted on a certain setof benchmark problems as demonstrated in [20]. More benchmarks can be foundin [19].

Allowing arbitrary shape for mesh elements provides a convenient flexibility inboth numerical approximation and mesh generation, especially in regions where thedomain geometry is complex. Such a flexibility is also very much appreciated inadaptive mesh refinement methods. This is a highly desirable feature in practicesince a single type of mesh technology is too restrictive in resolving complex multi-dimensional and multi-scale problems efficiently [21].

This paper is organized as follows. In Section 2, we present a discussion of weakdivergence operator in some weakly-defined spaces. In Section 3, we provide adetailed description and assumptions for the WG-MFEM. In Section 4, we definesome local projection operators and then derive some approximation propertieswhich are useful in error analysis. In Section 5, we shall establish an optimal ordererror estimate for the WG-MFEM approximations in a norm that is related to theL2 for the flux and H1 for the scalar function. In Section 6, we derive an optimalorder error estimate in L2 for the scalar approximation by using a duality argument




as was commonly employed in the standard Galerkin finite element methods [7,14].Finally, we provide some technical results in the appendix that are critical in dealingwith finite element functions on arbitrary polygons/polyhedra.

2. Weak divergence

The key in weak Galerkin methods is the use of weak derivatives in the placeof strong derivatives in the variational form for the underlying partial differentialequations. For the mixed problem (1.1) with boundary condition (1.2), the cor-responding variational form is given by (1.3) and (1.4), where divergence is theonly differential operator involved in the formulation. Thus, understanding weakdivergence is critically important in the corresponding WG method. The goal ofthis section is to introduce a weak divergence operator and its approximation byusing piecewise polynomials.

LetK be any polygonal or polyhedral domain with boundary ∂K. A weak vector-valued function on the region K refers to a vector-valued function v = {v0,vb} such

that v0 ∈ [L2(K)]d and vb ·n ∈ H− 12 (∂K), where n is the outward normal direction

of K on its boundary. The first component v0 can be understood as the value of vin the interior of K, and the second component vb represents v on the boundary ofK. The requirement of vb · n ∈ H− 1

2 (∂K) indicates that we are merely interestedin the normal component of vb. Note that vb may not be necessarily related to thetrace of v0 on ∂K should a trace be well defined. Denote by M(K) the space ofweak vector-valued functions on K; i.e.,

(2.1) M(K) = {v = {v0,vb} : v0 ∈ [L2(K)]d, vb · n ∈ H− 12 (∂K)}.

Following the definition of weak gradient introduced in [29], we define a weak di-vergence operator as follows.

Definition 2.1. The dual of L2(K) can be identified with itself by using thestandard L2 inner product as the action of linear functionals. With a similarinterpretation, for any v ∈ M(K), the weak divergence of v is defined as a linearfunctional ∇w · v in the dual space of H1(K) whose action on each ϕ ∈ H1(K) isgiven by

(2.2) (∇w · v, ϕ)K := −(v0,∇ϕ)K + 〈vb · n, ϕ〉∂K ,

where n is the outward normal direction to ∂K, (v0,∇ϕ)K =∫Kv0 · ∇ϕdK is the

action of v0 on ∇ϕ, and 〈vb · n, ϕ〉∂K is the action of vb · n on ϕ ∈ H12 (∂K).

The Sobolev space [H1(K)]d can be embedded into the space M(K) by aninclusion map iM : [H1(K)]d → M(K) defined as follows:

iM(q) = {q|K ,q|∂K}.With the help of the inclusion map iM, the Sobolev space [H1(K)]d can be viewedas a subspace of M(K) by identifying each q ∈ [H1(K)]d with iM(q). Analogously,a weak vector-valued function v = {v0,vb} ∈ M(K) is said to be in [H1(K)]d if itcan be identified with a function q ∈ [H1(K)]d through the above inclusion map.It is not hard to see that ∇w · v = ∇ · v if v is a smooth function in [H1(K)]d.

Next, we introduce a discrete weak divergence operator by approximating (∇w · )in a polynomial subspace of the dual of H1(K). To this end, for any non-negativeinteger r ≥ 0, denote by Pr(K) the set of polynomials on K with degree no more




xe

Ae

A

B

C

D

EF

n

Figure 1. Depiction of a shape-regular polygonal element ABCDEFA.

than r. A discrete weak divergence operator, denoted by (∇w,r·), is defined as theunique polynomial (∇w,r · v) ∈ Pr(K) that satisfies the following equation:

(2.3) (∇w,r · v, φ)K = −(v0,∇φ)K + 〈vb · n, φ〉∂K , ∀φ ∈ Pr(K).

3. Weak Galerkin MFEM: Assumptions and Algorithm

Let Th be a partition of the domain Ω consisting of polygons in 2D and polyhedrain 3D. Denote by Eh the set of all edges or flat faces in Th, and let E0

h = Eh\∂Ω bethe set of all interior edges or flat faces. For every element T ∈ Th, we denote by|T | the area or volume of T and by hT its diameter. Similarly, we denote by |e| thelength or area of e and by he the diameter of edge or flat face e ∈ Eh. We also setas usual the mesh size of Th by

h = maxT∈Th

hT .

All the elements of Th are assumed to be closed and simply connected polygonsor polyhedra; see Figure 1. We need some shape regularity for the partition Thdescribed as below.

A1: Assume that there exist two positive constants �v and �e such that forevery element T ∈ Th we have

(3.1) �vhdT ≤ |T |, �eh

d−1e ≤ |e|

for all edges or flat faces of T .A2: Assume that there exists a positive constant κ such that for every ele-

ment T ∈ Th we have

(3.2) κhT ≤ he

for all edges or flat faces e of T .A3: Assume that the mesh edges or faces are flat. We further assume that

for every T ∈ Th, and for every edge/face e ∈ ∂T , there exists a pyramidP (e, T,Ae) contained in T such that its base is identical with e, its apex isAe ∈ T , and its height is proportional to hT with a proportionality constant




σe bounded away from a fixed positive number σ∗ from below. In otherwords, the height of the pyramid is given by σehT such that σe ≥ σ∗ > 0.The pyramid is also assumed to stand up above the base e in the sensethat the angle between the vector xe−Ae, for any xe ∈ e, and the outwardnormal direction of e (i.e., the vector n in Figure 1) is strictly acute byfalling into an interval [0, θ0] with θ0 < π

2 .A4: Assume that each T ∈ Th has a circumscribed simplex S(T ) that is

shape regular and has a diameter hS(T ) proportional to the diameter ofT ; i.e., hS(T ) ≤ γ∗hT with a constant γ∗ independent of T . Furthermore,assume that each circumscribed simplex S(T ) intersects with only a fixedand small number of such simplices for all other elements T ∈ Th.

Figure 1 depicts a polygonal element that is shape regular. Readers are referredto [12] for a similar, but different type of shape regularity assumption for theunderlying finite element partition of the domain. The shape regularity is requiredfor deriving error estimates for locally defined projection operators to be detailedin later sections.

Recall that, on each element T ∈ Th, there is a space of weak vector-valuedfunctions M(T ) defined as in (6.11). Denote by M the weak vector-valued functionspace on Th given by

(3.3) M :=∏

T∈Th

M(T ).

Note that functions in M are defined on each element, and there are two-sidedvalues of vb on each interior edge/face e, depicted as vb|∂T1

and vb|∂T2in Figure 2.

ne

n1

n2

eT1

T2

A B

C

D

vb|∂T1vb|∂T2

Figure 2. An interior edge shared by two elements

Let V ⊂ M be a subspace of M consisting of weak vector-valued functions whichare continuous across each interior edge/face e in the normal direction; i.e.,

(3.4) V = {v ∈ M : ∃ q ∈ H(div,Ω) s.t. q|∂T · n = (vb)∂T · n, ∀T ∈ Th} .

The weak divergence operator as defined in (2.2) can be extended to any weakvector-valued function v ∈ V by taking weak divergence locally on each element T .




More precisely, the weak divergence of any v ∈ V is defined element-by-element asfollows:

∇w · v = ∇w · (v|T ), on T ∈ Th.Similarly, the discrete weak divergence as defined in (2.3) can be extended to V bydefining

(3.5) ∇w,r · v = ∇w,r · (v|T ), on T ∈ Th.The definition of weak divergence of v = {v0,vb} ∈ V requires the value of v on

each element T , namely v0, and the normal component of vb on each edge or facee ∈ Eh. Thus, it is the normal component of v on each e ∈ Eh that really enters intothe equation of discussion in numerical methods. For convenience, we introduce aset of normal directions on Eh as follows:

(3.6) Dh = {ne : ne is unit and normal to e, e ∈ Eh}.Figure 2 shows one example of ne pointing from the element T1 to T2. In therest of this paper, we will be concerned with a subspace of V in which the secondcomponent of v = {v0,vb} represents the normal component of v on each e ∈ Eh;i.e., (vb)|e = (v|e · ne)ne.

A discrete weak vector-valued function v = {v0, vb} refers to weak vector-valuedfunctions in which both v0 and vb are vector-valued polynomials. Since the secondcomponent is represented as vb = vbne where ne is the prescribed normal directionto e ∈ Eh, then vb is required to be a polynomial on each edge or flat face of ∂T .Recall that, for each element T ∈ Th, Pk(T ) denotes the set of polynomials on Twith degree no more than k ≥ 0 and P�(e) is the set of polynomials on e ∈ Eh withdegree no more than � ≥ 0.

We now introduce two finite element spaces which are necessary for formulatingour numerical schemes. The first one corresponds to the scalar (or pressure) variabledefined as

Wh = {w ∈ L2(Ω); w|T ∈ Pk+1(T )},where k ≥ 0 is a non-negative integer. The second one corresponds to vector-valuedfunctions and their normal components on the set Eh of edges or flat faces, and isgiven by

(3.7) Vh ={v = {v0,vb} : v0|T ∈ [Pk(T )]

d,vb|e = vbne, vb ∈ Pk(e), e ∈ Eh}.

The pair Vh ×Wh forms a finite element approximation space for the unknowns qand u of the problem (1.3)-(1.4). For simplicity of notation and discussion, we shallrefer to the above defined finite element spaces as the ([Pk(T )]

d, Pk(e), Pk+1(T ))elements. The lowest order of such elements makes use of piecewise constant forthe flux variable on each element T and its edges/faces and piecewise linear for thepressure (scalar) variable.

The discrete weak divergence (∇w,k+1·) as defined in (3.5) and (2.3) then pro-vides a linear map from the finite element space Vh to Wh. In particular, for anyv ∈ Vh and w ∈ Wh, we have the following relation:

(∇w,k+1 · v, w) : =∑T∈Th

(∇w,k+1 · v, w)T

=∑T∈Th

(∫∂T

(vb · n)wds−∫T

v0 · (∇w)dT

).(3.8)




With an abuse of notation, we shall use (∇w·) to denote the discrete weak divergenceoperator (∇w,k+1·) in the rest of this paper.

With the discrete divergence given by (3.8), one might naively formulate a finiteelement method by using (1.3) and (1.4) as follows. Find qh = {q0,qb} ∈ Vh anduh ∈ Wh such that

(αq0,v0)− (∇w · v, uh) = 〈g,vb · n〉∂Ω, ∀v = {v0,vb} ∈ Vh,(3.9)

(∇w · qh, w) = (f, w), ∀w ∈ Wh.(3.10)

Unfortunately, due to an insufficient enforcement on the component qb, the result-ing system of linear equations from (3.9)-(3.10) generally does not have a uniquesolution. One remedy to this problem is to stabilize the bilinear form (αq0,v0) byrequiring some communication between q0 and qb. To this end, we introduce threebilinear forms as follows.

a(η,v) = (αη0,v0), η,v ∈ Vh,(3.11)

b(η, w) = (∇w · η, w), η ∈ Vh, w ∈ Wh,(3.12)

s(η,v) = ρ∑T∈Th

hT 〈(η0 − ηb) · n, (v0 − vb) · n〉∂T , η,v ∈ Vh,(3.13)

and a stabilized bilinear form as(·, ·):

as(η,v) := a(η,v) + s(η,v).

Here ρ > 0 is any parameter and hT is the size of T . In practical computation, onemight chose ρ = 1 and substitute hT by the mesh size h for quasi-uniform partitions;i.e., partitions for which hT /h is bounded from below and above uniformly in T .

Weak Galerkin MFEM Algorithm. Let Th be a shape regular finite elementpartition of Ω and Vh×Wh be the corresponding finite element spaces consisting of([Pk(T )]

d, Pk(e), Pk+1(T )) elements with k ≥ 0. An approximation for (1.1)-(1.2)is given by seeking qh = {q0,qb} ∈ Vh and uh ∈ Wh such that

as(qh,v)− b(v, uh) = 〈g, vb · n〉∂Ω,(3.14)

b(qh, w) = (f, w),(3.15)

for all v = {v0,vb} ∈ Vh and w ∈ Wh. The pair of solutions (qh;uh) is called aweak Galerkin mixed finite element approximation of (1.1)-(1.2).

The WG-MFEM scheme (3.14)-(3.15) retains the mass conservation propertyof (1.1)-(1.2) locally on each element. This can be seen by choosing a piecewiseconstant test function w in (3.15). More precisely, for any element T ∈ Th, let wassume the value 1 on T and zero elsewhere. It follows from (3.15) that

(∇w · qh, 1)T = (f, 1)T .

The definition of discrete weak divergence (3.8) then implies the identity∫∂T

qb · nds =∫T

fdx,

which asserts a mass conservation for the WG-MFEM method.The rest of the paper shall provide a theoretical foundation for the solvability

and accuracy of the weak Galerkin mixed finite element scheme (3.14)-(3.15).




4. Local Projections: Definition and Properties

We introduce two projection operators into the finite element spaces Vh andWh by using local L2 projections. To this end, for each element T ∈ Th, denoteby Q0 the usual L2 projection from L2(T ) onto Pk(T ). Similarly, for each edgeor flat face e ∈ Eh, let Qb be the L2 projection from L2(e) onto Pk(e). For anyv = {v0, vbne} ∈ V with vb ∈ L2(e) on each edge/face e, denote by

Qhv := {Q0(v0), Qb(vb)ne}

the L2 projection of v in Vh. Next, denote by Qh the L2 projection from L2(Ω)onto Wh. Observe that Qh is in fact a composition of locally defined L2 projectionsinto the polynomial space Pk+1(T ) for each element T ∈ Th.

In the usual finite element error analysis, one often reduces the error for finiteelement solutions into the error between the exact solution and an appropriately de-fined local projection or interpolation of the solution. For the WG-MFEM methoddiscussed in previous sections, this refers to the error between the exact solutionand its local L2 projection. The difficulty in estimating the projection error arisesfrom the fact that the finite element partition Th contains arbitrary polygons orpolyhedra that are different from the usual simplices as commonly employed in thestandard finite element methods [14].

For simplicity of notation, we shall use � to denote less than or equal to up toa constant independent of the mesh size, variables, or other parameters appearingin the inequality.

Lemma 4.1. Let Th be a finite element partition of Ω satisfying the shape regularityassumptions A1–A4 as given in Section 3. Then, we have∑

T∈Th‖q−Q0q‖2T � h2(s+1)‖q‖2s+1, 0 ≤ s ≤ k,(4.1) ∑

T∈Th‖∇(q−Q0q)‖2T � h2s‖q‖2s+1, 0 ≤ s ≤ k,(4.2) ∑

T∈Th

(‖u−Qhu‖2T + h2

T ‖∇(u−Qhu)‖2T)

(4.3)

� h2(s+1)‖u‖2s+1, 0 ≤ s ≤ k + 1.

Proof. To derive (4.1), let S(T ) be the circumscribed simplex of T on which q can

be defined by smooth extension if necessary. Let Q0q be the projection of q in theelement defined on S(T ). It follows that

(4.4) ‖q−Q0q‖T ≤ ‖q− Q0q‖T ≤ ‖q− Q0q‖S(T ) � hs+1T ‖q‖s+1,S(T ).

Using the above estimate we obtain

(4.5)∑T∈Th

‖q−Q0q‖2T � h2(s+1)∑T∈Th

‖q‖2s+1,S(T ).

It follows from the assumption A4 that the set of the circumscribed simplices{S(T ) : T ∈ Th} has a fixed and small number of overlaps. Thus, the followingestimate holds true: ∑

T∈Th

‖q‖2s+1,S(T ) � ‖q‖2s+1.

Substituting the above inequality into (4.5) yields the desired estimate (4.1).




To derive (4.2), we use the triangle inequality and the standard error estimateon S(T ) to obtain

‖∇(q−Q0q)‖T ≤ ‖∇(q− Q0q)‖S(T ) + ‖∇(Q0q−Q0q)‖S(T )

� hsT ‖q‖s+1,S(T ) + h−1

T ‖Q0q−Q0q‖S(T ),(4.6)

where we have also used the standard inverse inequality in the second line. Noticethat the assumption A3 on Th implies that there exists a ball B ⊂ T with adiameter proportional to hT . Thus, we have from Lemma A.5 (see Appendix) that

‖Q0q−Q0q‖S(T ) � ‖Q0q−Q0q‖B ≤ ‖Q0q−Q0q‖T≤ ‖q− Q0q‖T + ‖q−Q0q‖T� hs+1

T ‖q‖s+1,S(T ).

Substituting the above estimate into (4.6) yields

‖∇(q−Q0q)‖T � hsT ‖q‖s+1,S(T ).

Summing up the above estimate over T ∈ Th leads to the desired estimate (4.2).Finally, the estimate (4.3) can be established analogously to (4.1) and (4.2). �

We shall derive two equations that play useful roles in the error analysis for theWG-MFEM. The first equation is given by the following lemma.

Lemma 4.2. For any q ∈ [H1(Ω)]d, let Qhq ∈ Vh be the projection given by localL2 projections. Then, on each element T , we have

(∇w · (Qhq), w)T = (∇ · q, w)T − 〈q · n−Qb(q · n), w〉∂T(4.7)

for all w ∈ Pk+1(T ). Moreover, by summing (4.7) over all T ∈ Th we obtain

b(Qhq, w) = (∇ · q, w)−∑T∈Th

〈q · n−Qb(q · n), w〉∂T .(4.8)

Proof. For q ∈ [H1(Ω)]d, we apply the definition of Qh and the weak divergence∇w · (Qhq) to obtain

(∇w ·Qhq, w)T = −(Q0q, ∇w)T + 〈Qb(q · ne)ne · n, w〉∂T= −(Q0q, ∇w)T + 〈Qb(q · n), w〉∂T(4.9)

for all w ∈ Pk+1(T ). Here we have used the fact that ne = ±n. Since Q0 is the L2

projection into [Pk(T )]d, then

(Q0q, ∇w)T = (q,∇w)T

= −(∇ · q, w)T + 〈q · n, w〉∂T ,(4.10)

where we have applied the usual divergence theorem in the second line. Substituting(4.10) into (4.9) yields (4.7). This completes the proof of the lemma. �

The second equation is concerned with the bilinear form (∇w ·v,Qhw) for v ∈ Vh

and w ∈ H1(Ω). Using the definition of Qh and the integration by parts, we have




for any v = {v0,vb} ∈ Vh and w ∈ H1(T ) that

(∇w · v, Qhw)T = −(v0, ∇(Qhw))T + 〈vb · n, Qhw〉∂T= (∇ · v0, Qhw)T + 〈(vb − v0) · n, Qhw〉∂T= (∇ · v0, w)T + 〈(vb − v0) · n, Qhw〉∂T= −(v0, ∇w)T + 〈v0 · n, w〉∂T + 〈(vb − v0) · n, Qhw〉∂T= −(v0, ∇w)T + 〈(v0 − vb) · n, w −Qhw〉∂T + 〈vb · n, w〉∂T .

The result can be summarized as follows.

Lemma 4.3. For any v = {v0,vb} ∈ Vh and w ∈ H1(T ) on the element T ∈ Th,one has

(∇w · v, Qhw)T = −(v0, ∇w)T + 〈(v0 − vb) · n, w −Qhw〉∂T+〈vb · n, w〉∂T .(4.11)

Moreover, if w ∈ H1(Ω), then the sum of (4.11) over all T ∈ Th gives

b(v, Qhw) = −(v0, ∇hw) +∑T∈Th

〈(v0 − vb) · n, w −Qhw〉∂T

+〈vb · n, w〉∂Ω,(4.12)

where ∇hw is the gradient of w taken element-by-element.

Let T ∈ Th be any element, and q ∈ [H1(T )]d. Then, we have from the definitionof Qb that

‖q · n−Qb(q · n)‖2∂T = 〈q · n−Qb(q · n), q · n−Qb(q · n)〉∂T= 〈q · n−Qb(q · n), q · n− (Q0q) · n)〉∂T≤ ‖q · n−Qb(q · n)‖∂T ‖q · n− (Q0q) · n‖∂T .

It follows that

(4.13) ‖q · n−Qb(q · n)‖∂T ≤ ‖q · n− (Q0q) · n‖∂T .

Lemma 4.4. Let q ∈ [Hk+1(Ω)]d and s be any real number such that 0 ≤ s ≤ k.Then, we have

(4.14)∑T∈Th

hT ‖q · n−Qb(q · n)‖2∂T � h2(s+1)‖q‖2s+1

and

(4.15) |||q−Qhq||| � ρhs+1‖q‖s+1.

Here ||| · ||| is a norm in V defined by (5.7)

Proof. Apply the trace inequality (A.18) (see Appendix) to the right-hand side of(4.13) to obtain

(4.16) hT ‖q · n−Qb(q · n)‖2∂T � ‖q−Q0q‖2T + h2T ‖∇(q−Q0q)‖2T .

Thus, it follows from Lemma 4.1 that the estimate (4.14) holds true.




To derive (4.15), we have from the definition of the bilinear form s(·, ·) in (3.13),the definition of Qb, and the inequality (4.13) that

s(q−Qhq,q−Qhq) = ρ∑T∈Th

hT ‖(q−Q0q) · n− (q · n−Qb(q · n))‖2∂T

� ρ∑T∈Th

hT

(‖(q−Q0q) · n‖2∂T + ‖q · n−Qb(q · n)‖2∂T

)� ρ

∑T∈Th

hT ‖(q−Q0q) · n‖2∂T .

Now apply the trace inequality (A.18) to the last term of the above inequality toobtain

(4.17) s(q−Qhq,q−Qhq) � ρ∑T∈Th

(‖q−Q0q‖2T + h2

T ‖∇(q−Q0q)‖2T).

Finally, it follows from the triple-bar norm definition (5.7) and the above inequalitythat

|||q−Qhq|||2 = a(q−Qhq,q−Qhq) + s(q−Qhq,q−Qhq)

= ‖α 12 (q−Q0q)‖2 + s(q−Qhq,q−Qhq),

which, combined with Lemma 4.1, gives rise to the desired estimate (4.15). Thiscompletes the proof. �

5. Error analysis

The goal of this section is to derive some optimal order error estimates for theWG-MFEM approximation (qh;uh) obtained from (3.14)-(3.15). In finite elementanalysis, it is routine to decompose the error into two components in which thefirst is the difference between the exact solution and a properly defined projectionof the exact solution while the second is the difference of the finite element solutionwith the same projection. In the current application, we shall employ the followingdecomposition:

q− qh = (q−Qhq) + (Qhq− qh),

u− uh = (u−Qhu) + (Qhu− uh).

For simplicity, we introduce the following notation:

eh = {e0, eb} := qh −Qhq, εh := uh −Qhu.

Moreover, we shall denote∑

T∈Th‖∇w‖2T by ‖∇hw‖2 for any w ∈

∏T∈Th

H1(T ).

5.1. Error equations. The goal here is to derive equations that eh and εh mustsatisfy. To this end, for any finite element function v = {v0,vb} ∈ Vh, we test thefirst equation of (1.1) against v0 to obtain

(αq,v0) + (∇u,v0) = 0.

Now applying the identity (4.12) to the term (∇u,v0) yields

a(q,v)− b(v,Qhu) =∑T∈Th

〈(v0 − vb) · n,Qhu− u〉∂T + 〈g,vb · n〉∂Ω,




where the Dirichlet boundary condition (1.2) was also used. Recall that as(·, ·) =a(·, ·) + s(·, ·). Adding and subtracting as(Qhq,v) in the equation above gives

as(Qhq,v)− b(v,Qhu) = a(Qhq− q, v) + s(Qhq,v) + 〈g,vb · n〉∂Ω+

∑T∈Th

〈(v0 − vb) · n,Qhu− u〉∂T .(5.1)

It follows from the definition of the bilinear form s(·, ·) that, for any w ∈ [H1(Ω)]d,one has

(5.2) s(w,v) = 0, ∀v ∈ V .Thus, we have s(Qhq,v) = s(Qhq− q,v) and

a(Qhq− q, v) + s(Qhq,v) = as(Qhq− q,v).

Substituting the above into (5.1) yields

as(Qhq,v)− b(v,Qhu) = as(Qhq− q, v) + 〈g,vb · n〉∂Ω+

∑T∈Th

〈(v0 − vb) · n,Qhu− u〉∂T .(5.3)

Next, we test the second equation of (1.1) against any w ∈ Wh to obtain

(∇ · q, w) = (f, w).

Substituting (4.8) into the above equation yields

(5.4) b(Qhq, w) = (f, w)−∑T∈Th

〈q · n−Qb(q · n), w〉∂T .

Now, subtracting (5.3) from (3.14) gives

as(eh,v)− b(v, εh) = as(q−Qhq, v)

−∑T∈Th

〈(v0 − vb) · n, Qhu− u〉∂T ,(5.5)

for all v ∈ Vh. Similarly, subtracting (5.4) from (3.15) yields

(5.6) b(eh, w) =∑T∈Th

〈q · n−Qb(q · n), w〉∂T ,

for all w ∈ Wh.The equations (5.5) and (5.6) constitute governing rules for the error term eh

and εh. These are called error equations.

5.2. Boundedness and inf-sup condition. For any interior edge or flat facee ∈ Eh, let T1 and T2 be two elements sharing e in common. The jump of w ∈ Wh

on e is given by

[[w]]e =

{w|

∂T1− w|

∂T2, e ∈ E0

h,w, e ∈ ∂Ω.

We introduce a norm in V as follows:

(5.7) |||v|||2 := a(v,v) + s(v,v), v ∈ V .Furthermore, let ‖ · ‖1,h be a norm in the space Wh +H1(Ω) defined as follows:

(5.8) ‖w‖21,h :=∑T∈Th

‖∇w‖2T +∑e∈Eh

h−1e ‖Qb[[w]]‖2e.




It is not hard to see that ‖ · ‖1,h defines a norm in Wh since ‖w‖1,h = 0 would leadto w = const on each element T and [[w]] = Qb[[w]] = 0 on each edge or flat facee ∈ Eh. It follows that w = 0 on each element T . As to ||| · |||, assume that |||v||| = 0for some v ∈ Vh; i.e.,

(αv0,v0) + ρ∑T∈Th

hT 〈(v0 − vb) · n, (v0 − vb) · n〉∂T = 0.

This implies that v0 = 0 on each element T and (v0 − vb) · n = 0 on each edge orflat face e ∈ Eh. Thus, we have vb ·n = 0 on each e ∈ Eh. Recall that vb is a vectorparallel to n on each e ∈ Eh. Hence, we must have vb = 0 on each e ∈ Eh. Theother properties for a norm can be easily verified for ‖ · ‖1,h and ||| · |||.

It is not hard to see that the triple-bar norm is essentially a discrete L2 norm,and the norm ‖ · ‖1,h is an H1-equivalent norm in the corresponding space.

Lemma 5.1. For any ϕ ∈ Wh, there exists at least one v ∈ Vh such that

|b(v, ϕ)| = ‖ϕ‖21,h,(5.9)

|||v||| � ‖ϕ‖1,h.(5.10)

Therefore, the following inf-sup condition is satisfied:

(5.11) supv∈Vh, |||v|||�=0

|b(v, ϕ)||||v||| ≥ β‖ϕ‖1,h

for some positive constant β independent of the meshsize h.

Proof. It follows from the definition of b(·, ·) and the discrete divergence that

b(v, ϕ) = (∇w · v, ϕ)=

∑T∈Th

(〈vb · n, ϕ〉∂T − (v0, ∇ϕ)T )

=∑e∈Eh

〈vb · ne, [[ϕ]]〉e −∑T∈Th

(v0, ∇ϕ)T ,

provided that the jump [[ϕ]] was taken consistently with the direction set Dh. Nowby taking v0 = −∇ϕ on each element T and vb = h−1

e Qb([[ϕ]])ne on each edge orflat face e we arrive at

b(v, ϕ) =∑T∈Th

(∇ϕ, ∇ϕ)T +∑e∈Eh

h−1e 〈Qb[[ϕ]], Qb[[ϕ]]〉e = ‖ϕ‖21,h,

which verifies (5.9). To verify (5.10), we apply the definition of the triple-bar normto the above chosen v to obtain

|||v|||2 = (α∇hϕ,∇hϕ) + ρ∑T∈Th

hT ‖∇ϕ · n+ h−1e Qb([[ϕ]])ne · n‖2∂T

� ‖∇hϕ‖2 +∑e∈Eh

h−1e ‖Qb[[ϕ]]‖2e = ‖ϕ‖21,h,

which is what (5.10) states. �

Lemma 5.2. The bilinear form b(·, ·) is bounded in Vh×Wh. In other words, thereexists a constant C such that

(5.12) |b(v, ϕ)| ≤ C|||v||| ‖ϕ‖1,h.




Proof. We note from the definition of weak divergence that

b(v, ϕ) = (∇w · v, ϕ)=

∑T∈Th

(〈vb · n, ϕ〉∂T − (v0,∇ϕ)T )

=∑e∈Eh

〈vb · ne, [[ϕ]]〉e −∑T∈Th

(v0,∇ϕ)T .

Using Cauchy-Schwarz we obtain

|b(v, ϕ)| ≤∑e∈Eh

‖vb · ne‖e ‖[[ϕ]]‖e +∑T∈Th

‖v0‖T ‖∇ϕ‖T

≤(∑

e∈Eh

he‖vb · ne‖2e

) 12(∑

e∈Eh

h−1e ‖[[ϕ]]‖2e

) 12

+ ‖v0‖ ‖∇hϕ‖

≤

⎛⎝(∑

e∈Eh

he‖vb · ne‖2e

) 12

+ ‖v0‖

⎞⎠ ‖ϕ‖1,h.(5.13)

Let T be an element that takes e as an edge or flat face. Then, using the traceinequality (A.18) and the inverse inequality (A.38) we obtain

he‖vb · ne‖2e ≤ 2he‖(vb − v0) · ne‖2e + 2he‖v0 · ne‖2e� he‖(vb − v0) · ne‖2e + ‖v0‖2T .

Substituting the above inequality into (5.13) yields

|b(v, ϕ)| � |||v||| ‖ϕ‖1,h,

which completes the proof of the lemma. �

5.3. Error estimates. We first establish some estimates useful in the forthcomingerror estimates.

Lemma 5.3. Let u ∈ Hk+2(Ω) and q ∈ [Hk+1(Ω)]d be two smooth functions onΩ. Then, the following estimates hold true:

∑T∈Th

|〈(v0 − vb) · n, Qhu− u〉∂T | � hs+1 ‖u‖s+2|||v|||, 0 ≤ s ≤ k,(5.14)

∑T∈Th

|〈q · n−Qb(q · n), ϕ〉∂T | � hs+1 ‖q‖s+1‖∇hϕ‖, 0 ≤ s ≤ k,(5.15)

for all v ∈ V and ϕ ∈∏

T∈ThH1(T ).




Proof. It follows from the Cauchy-Schwarz inequality, the definition of ||| · |||, andthe trace inequality (A.18) that∑

T∈Th

|〈(v0 − vb) · n, Qhu− u〉∂T |

�( ∑

T∈Th

hT ‖(v0 − vb) · n‖2∂T

) 12

·( ∑

T∈Th

h−1T ‖Qhu− u‖2∂T

) 12

� |||v|||( ∑

T∈Th

(h−2T ‖Qhu− u‖2T + ‖∇(Qhu− u)‖2T

)) 12

� hs+1|||v||| ‖u‖s+2,

where we have used the estimate (4.3). This verifies the validity of (5.14).Next, let ϕ be the average of ϕ over each element T . It follows from the definition

of Qb, the estimates (4.14), and (A.18) that∑T∈Th

|〈q · n−Qb(q · n), ϕ〉∂T | =∑T∈Th

|〈q · n−Qb(q · n), ϕ− ϕ〉∂T |

≤( ∑

T∈Th

hT ‖q · n−Qb(q · n)‖2∂T

) 12( ∑

T∈Th

h−1T ‖ϕ− ϕ‖2∂T

) 12

� hs+1‖q‖s+1

( ∑T∈Th

(h−2T ‖ϕ− ϕ‖2T + ‖∇ϕ‖2T

)) 12

� hs+1‖q‖s+1‖∇hϕ‖,which completes the proof of (5.15). �

With the help of Lemma 5.3, Lemma 5.1, and Lemma 5.2, we are now in aposition to derive an error estimate for the WG-MFEM solution (qh;uh) in thenorm ||| · ||| and ‖ · ‖1,h, respectively.Theorem 5.4. Let qh ∈ Vh and uh ∈ Wh be the solution of the weak Galerkinmixed finite element scheme (3.14)-(3.15). Assume that the exact solution (q;u) of(1.3)-(1.4) is regular such that u ∈ Hs+2(Ω) and q ∈ [Hs+1(Ω)]d with s ∈ (0, k].Then, one has

(5.16) |||qh −Qhq|||+ ‖uh −Qhu‖1,h � hs+1 (‖u‖s+2 + ‖q‖s+1) .

Proof. Let

�(v) := as(q−Qhq, v)−∑T∈Th

〈(v0 − vb) · n, Qhu− u〉∂T

be a linear functional on the finite element space Vh. Also, let

χ(w) :=∑T∈Th

〈q · n−Qb(q · n), w〉∂T

be a linear functional on the finite element space Wh. The error equations (5.5)and (5.6) indicate that the pair (eh; εh) is a solution of the following problem:

as(eh,v)− b(v, εh) = �(v), ∀v ∈ Vh,(5.17)

b(eh, w) = χ(w), ∀w ∈ Wh.(5.18)




The bilinear form as(·, ·) is clearly bounded, symmetric and positive definite inVh equipped with the triple-bar norm ||| · |||. Lemma 5.2 indicates that the bilinearform b(·, ·) is bounded in Vh ×Wh, and Lemma 5.1 proved that the usual inf-supcondition of Babuska [3] and Brezzi [8] is satisfied. Thus, we have from the generaltheory of Babuska and Brezzi that

(5.19) |||eh|||+ ‖εh‖1,h ≤ C(‖�‖V′

h+ ‖χ‖W′

h

),

where ‖�‖V′hand ‖χ‖W′

hare the norm of the corresponding linear functionals. To

estimate the norms, we use (4.15) and (5.14) to come up with

|�(v)| � hs+1 (‖q‖s+1 + ‖u‖s+2) |||v|||.

Thus, we have

(5.20) ‖�‖V′h� hs+1 (‖q‖s+1 + ‖u‖s+2) , 0 ≤ s ≤ k.

As to the norm of χ, we use the estimate (5.15) to obtain

|χ(w)| � hs+1‖q‖s+1‖w‖1,h, 0 ≤ s ≤ k,

which implies that

(5.21) ‖χ‖W′h� hs+1‖q‖s+1, 0 ≤ s ≤ k.

Substituting (5.20) and (5.21) into (5.19) yields the desired error estimate (5.16).�

6. Error estimates in L2

To obtain an optimal order error estimate for the scalar component εh = uh−Qhuin the usual L2 norm, we consider a dual problem that seeks Ψ and φ satisfying

αΨ+∇φ = 0, in Ω,(6.1)

∇ ·Ψ = εh, in Ω,(6.2)

φ = 0, on ∂Ω.(6.3)

Assume that the usual H2-regularity is satisfied for the dual problem; i.e., for anyεh ∈ L2(Ω), there exists a unique solution (Ψ;φ) ∈ [H1(Ω)]d ×H2(Ω) such that

(6.4) ‖φ‖2 + ‖Ψ‖1 � ‖εh‖.

Lemma 6.1. Let (q;u) be the solution of (1.3)-(1.4). Assume that u ∈ Hk+2(Ω),q ∈ [Hk+1(Ω)]d, and the coefficient function satisfies α ∈ W 1,∞(T ) on each elementT . Then, one has the following estimates:

|as(q−Qhq, QhΨ)| � hk+2‖q‖k+1‖Ψ‖1,(6.5) ∑T∈Th

|〈Q0Ψ · n−Qb(Ψ · n),Qhu− u〉∂T | � hk+2‖u‖k+2‖Ψ‖1,(6.6)

∑T∈Th

|〈q · n−Qb(q · n),Qhφ− φ〉∂T | � hk+2‖q‖k+1‖φ‖2.(6.7)




Proof. Let α be the average of α on each element T . It follows from the definitionof as(·, ·) and the approximation property (4.15) that

|as(q−Qhq, QhΨ)| ≤ |(α(q−Q0q), Q0Ψ)|+ |s(q−Q0q, QhΨ)|= |(q−Q0q, (α− α)Q0Ψ)|+ |s(q−Q0q, QhΨ−Ψ)|≤ Ch‖∇α‖∞‖q−Q0q‖ ‖Q0Ψ‖+ |||q−Qhq||| |||Ψ−QhΨ|||� hk+2‖q‖k+1‖Ψ‖1,

where ‖∇α‖∞ is the L∞ norm of ∇α taken on each element. As to (6.6), we use(A.18) and (4.15) to obtain∑

T∈Th

|〈Q0Ψ · n−Qb(Ψ · n),Qhu− u〉∂T |

=∑T∈Th

|〈(Q0Ψ−Ψ) · n− (Qb(Ψ · n)−Ψ · n),Qhu− u〉∂T |

� |||QhΨ−Ψ|||( ∑

T∈Th

(h−2T ‖Qhu− u‖2T + ‖∇(Qhu− u)‖2T )

)1/2

� hk+2‖u‖k+2‖Ψ‖1.Similarly, using (4.14) we have∑

T∈Th

|〈q · n−Qb(q · n), Qhφ− φ〉∂T |

� hk+1‖q‖k+1

( ∑T∈Th

(h−2T ‖Qhφ− φ‖2T + ‖∇(Qhφ− φ)‖2T )

) 12

� hk+2‖q‖k+1‖φ‖2.This completes the proof of the lemma. �

We are now in a position to establish an optimal order error estimate for thescalar/pressure component εh = uh − Qhu in the usual L2-norm. The result isstated in the following theorem.

Theorem 6.2. Let qh ∈ Vh and uh ∈ Wh be the solution of (3.14)-(3.15). Assumethat the exact solution of (1.3)-(1.4) satisfies u ∈ Hk+2(Ω) and q ∈ [Hk+1(Ω)]d.Then, one has the following error estimate:

(6.8) ‖uh −Qhu‖ � hk+2 (‖u‖k+2 + ‖q‖k+1) .

Proof. Testing (6.2) by εh = uh −Qhu and then using (4.8) yields

(εh, εh) = (∇ ·Ψ, εh)

= b(QhΨ, εh) +∑T∈Th

〈Ψ · n−Qb(Ψ · n), εh〉∂T .(6.9)

It follows from (5.5) that

b(QhΨ, εh) = as(eh, QhΨ)− as(q−Qhq, QhΨ)

+∑T

〈(Q0Ψ) · n−Qb(Ψ · n), Qhu− u〉∂T .




Substituting the above equation into (6.9) yields

(εh, εh) = as(eh, QhΨ)− as(q−Qhq, QhΨ)

+∑T∈Th

〈Q0Ψ · n−Qb(Ψ · n), Qhu− u〉∂T(6.10)

+∑T∈Th

〈Ψ · n−Qb(Ψ · n), εh〉∂T .

Using (6.6) we obtain

(6.11)∑T∈Th

〈Q0Ψ · n−Qb(Ψ · n), Qhu− u〉∂T � hk+2‖u‖k+2 ‖Ψ‖1.

Using (5.15) we arrive at

(6.12)∑T∈Th

〈Ψ · n−Qb(Ψ · n), εh〉∂T � h‖∇hεh‖ ‖Ψ‖1.

From (6.5) we have

(6.13) |as(q−Qhq, QhΨ)| � hk+2‖q‖k+1 ‖Ψ‖1.Now substituting (6.11)-(6.13) into (6.10) we obtain

(6.14) (εh, εh) � |as(eh, QhΨ)|+hk+2 (‖q‖k+1 + ‖u‖k+2) ‖Ψ‖1+h‖∇hεh‖ ‖Ψ‖1.It remains to deal with |as(eh, QhΨ)| in (6.14). To this end, we note that

as(eh, QhΨ) = as(eh, QhΨ−Ψ) + as(eh,Ψ)

and|as(eh, QhΨ−Ψ)| ≤ |||eh||| |||QhΨ−Ψ||| � h|||eh||| ‖Ψ‖1.

Thus,

(6.15) |as(eh, QhΨ)| � |as(eh,Ψ)|+ h|||eh||| ‖Ψ‖1.Since Ψ ∈ [H1(Ω)]d, we have as(eh,Ψ) = a(eh,Ψ); i.e., the stabilization termvanishes when one of the components is sufficiently regular. Now testing (6.1)against eh gives

a(eh,Ψ) = (αΨ, e0) = −(∇φ, e0).

Furthermore, we have from (4.12) (with v = eh and w = φ) and the fact that φ = 0on ∂Ω that

(∇φ, e0) = −b(eh,Qhφ) +∑T∈Th

〈(e0 − eb) · n, φ−Qhφ〉∂T .

Using the error equation (5.6) to replace the term b(eh,Qhφ) in above equation weobtain

(∇φ, e0) = −∑T∈Th

〈q · n−Qb(q · n),Qhφ〉∂T +∑T∈Th

〈(e0 − eb) · n, φ−Qhφ〉∂T

=∑T∈Th

〈q · n−Qb(q · n), φ−Qhφ〉∂T +∑T∈Th

〈(e0 − eb) · n, φ−Qhφ〉∂T ,

where we have used the fact that φ ∈ H1(Ω) and φ = 0 on ∂Ω in the second line.Now using (6.7) to estimate the first summation in the above equation, and (5.14)with s = 0, v = eh, u = φ to estimate the second summation we obtain

|(∇φ, e0)| � (hk+2‖q‖k+1 + h|||eh|||) ‖φ‖2.




Thus,

|as(eh,Ψ)| = |a(eh,Ψ)| = |(∇φ, e0)| � (hk+2‖q‖k+1 + h|||eh|||) ‖φ‖2.Substituting the above into (6.15) yields

(6.16) |as(eh, QhΨ)| � (hk+2‖q‖k+1 + h|||eh|||) ‖φ‖2 + h|||eh||| ‖Ψ‖1.Now substituting (6.16) into (6.14) gives

(εh, εh) �(hk+2‖q‖k+1 + h|||eh|||) ‖φ‖2 + h|||eh||| ‖Ψ‖1(6.17)

+ hk+2 (‖q‖k+1 + ‖u‖k+2) ‖Ψ‖1 + h‖∇hεh‖ ‖Ψ‖1.

Observe that theH2-regularity of the dual problem implies the following estimate

‖φ‖2 + ‖Ψ‖1 � ‖εh‖.Substituting the above into (6.17) and then dividing both sides by ‖εh‖ we obtain

‖εh‖ � h(|||eh|||+ ‖∇hεh‖) + hk+2 (‖q‖k+1 + ‖u‖k+2) ,

which, together with (5.16), implies the desired L2 error estimate (6.8). This com-pletes the proof. �

Appendix

The goal of this appendix is to establish some fundamental estimates useful inthe error estimate for WG-MFEM. First, we derive a trace inequality for functionsdefined on the finite element partition Th with properties specified in Section 3.

Lemma A.3 (Trace Inequality). Let Th be a partition of the domain Ω into poly-gons in 2D or polyhedra in 3D. Assume that the partition Th satisfies the assump-tions (A1), (A2), and (A3) as specified in Section 3. Then, there exists a constantC such that for any T ∈ Th and edge/face e ∈ ∂T , we have

(A.18) ‖θ‖2e ≤ C(h−1T ‖θ‖2T + hT ‖∇θ‖2T

),

where θ ∈ H1(T ) is any function.

Proof. We shall provide a proof for the case of e being a flat face and θ ∈ C1(T ).To this end, let the flat face e be represented by the following parametric equation:

xe = φ(ξ, η) := (φ1(ξ, η), φ2(ξ, η), φ3(ξ, η))(A.19)

for (ξ, η) ∈ D ⊂ R2. By Assumption A3, there exists a pyramid P (e, T,Ae) withapex Ae = x∗ := (x∗

1, x∗2, x

∗3) contained in the element T ; see Figure 1. This pyramid

has the following parametric representation:

x(t, ξ, η) = (1− t)φ(ξ, η) + tx∗(A.20)

for (t, ξ, η) ∈ [0, 1] ×D. For any given xe ∈ e, the line segment joining xe and theapex x∗ can be represented by

x(t) = xe + t(x∗ − xe).

From the fundamental theorem of Calculus, we have

θ2(xe)− θ2(x(t)) = −∫ t

0

∂τθ2(xe + τω)dτ, ω = x∗ − xe.




The above can be further rewritten as

θ2(xe)− θ2(x(t)) = −2

∫ t

0

θ (∇θ · ω) dτ.

It follows from the Cauchy-Schwarz inequality that for any t ∈ [0, 12 ] we have

θ2(xe) ≤ θ2(x(t)) + h−1T

∫ 12

0

θ2|ω|dτ + hT

∫ 12

0

|∇θ|2|ω|dτ.

Now we integrate the above inequality over the flat face e by using the parametricequation (A.19), yielding∫

D

θ2(xe)|φξ × φη|dξdη ≤∫D

θ2(x(t))|φξ × φη|dξdη

+ h−1T

∫ 12

0

∫D

θ2|ω||φξ × φη|dξdηdτ(A.21)

+ hT

∫ 12

0

∫D

|∇θ|2|ω||φξ × φη|dξdηdτ.(A.22)

Observe that the integral of a function over the prismatoid

P 12:= {x(t, ξ, η) : (t, ξ, η) ∈ [0, 1/2]×D}

is given by ∫P 1

2

f(x)dx =

∫ 12

0

∫D

f(x(τ, ξ, η))J(τ, ξ, η)dξdηdτ,

where J(τ, ξ, η) = (1−τ )2|(φξ×φη) ·ω| is the Jacobian from the coordinate change.The vector φξ×φη is normal to the face e, and ω = x∗−xe is a vector from the basepoint xe to the apex x∗. The angle assumption (see Assumption A3 of Section 3)for the prism P (e, T,Ae) indicates that the Jacobian satisfies the relation

(A.23) J(τ, ξ, η) ≥ μ0

4|φξ × φη| |ω|, τ ∈ [0, 1/2]

for some fixed μ0 ∈ (0, 1). Observe that the left-hand side of (A.21) is the surfaceintegral of θ2 over the face e. Thus, substituting (A.23) into (A.21) and (A.22)yields ∫

e

θ2de ≤∫D

θ2(x(t))|φξ × φη|dξdη

+4μ−10 h−1

T

∫P 1

2

θ2dx+ 4μ−10 hT

∫P 1

2

|∇θ|2dx.

Now we integrate the above with respect to t in the interval [0, 12 ] to obtain

1

2

∫e

θ2de ≤∫ 1

2

0

∫D

θ2(x(t))|φξ × φη|dξdηdt(A.24)

+2μ−10 h−1

T

∫P 1

2

θ2dx+ 2μ−10 hT

∫P 1

2

|∇θ|2dx.




Again, by substituting (A.23) into the right-hand side of (A.24) we arrive at

1

2

∫e

θ2de ≤ 4μ−10 |ω|−1

∫ 12

0

∫D

θ2(x(t))J(t, ξ, η)dξdηdt(A.25)

+2μ−10 h−1

T

∫P 1

2

θ2dx+ 2μ−10 hT

∫P 1

2

|∇θ|2dx.

The integral on the right-hand side of (A.25) is the integral of θ2 on the prismatoidP 1

2. It can be seen from the Assumption A3 that

|ω|−1 ≤ α∗h−1T

for some positive constant α∗. Therefore, we have from the above estimate that∫e

θ2de ≤ Ch−1T

∫P 1

2

θ2dx+ ChT

∫P 1

2

|∇θ|2dx,

which completes the proof of the lemma. �

Next, we would like to establish an estimate for the L2 norm of polynomialfunctions by their L2 norm on a subdomain. To this end, we first derive a result ofsimilar nature for general functions in H1.

Lemma A.4. Let K ⊂ Rd be convex and v ∈ H1(K). Then,

(A.26) ‖v‖2K ≤ 2|K||S| ‖v‖2S +

4δ2(d+1)

|S|2d2 ‖∇v‖2K ,

where δ is the diameter of K and S is any measurable subset of K.

Proof. Since C1(K) is dense in H1(K), it is sufficient to establish (A.26) for v ∈C1(K). For any x,y ∈ K, we have

v2(x) = v2(y)−∫ |x−y|

0

∂rv2(x+ rω)dr, ω =

y − x

|y − x| .

From the usual chain rule and the Cauchy-Schwarz inequality we obtain

v2(x) = v2(y)− 2

∫ |x−y|

0

v∂rv(x+ rω)dr

≤ v2(y) + εδ−1

∫ |x−y|

0

v2dr + ε−1δ

∫ |x−y|

0

|∂rv|2dr,(A.27)

where ε > 0 is any constant. Let

V (x) =

{v2(x), x ∈ K,0, x /∈ K

and

W (x) =

{|∂rv|2(x), x ∈ K,0, x /∈ K.

Then, the inequality (A.27) can be rewritten as

v2(x) ≤ v2(y) + εδ−1

∫ ∞

0

V (x+ rω)dr + ε−1δ

∫ ∞

0

W (x+ rω)dr.




Integrating the above inequality with respect to y in S yields

|S|v2(x) ≤∫S

v2dS(A.28)

+

∫|x−y|≤δ

(εδ−1

∫ ∞

0

V (x+ rω)dr + ε−1δ

∫ ∞

0

W (x+ rω)dr

)dy.

It is not hard to see that∫|x−y|≤δ

∫ ∞

0

V (x+ rω)drdy =

∫ ∞

0

∫|ω|=1

∫ δ

0

V (x+ rω)ρd−1 dρ dω dr

=δd

d

∫ ∞

0

∫|ω|=1

V (x+ rω)dω dr

=δd

d

∫K

|x− y|1−dv2(y)dy.(A.29)

Analogously, we have

(A.30)

∫|x−y|≤δ

∫ ∞

0

W (x+ rω)drdy =δd

d

∫K

|x− y|1−d|∂rv(y)|2dy.

Substituting (A.29) and (A.30) into (A.28) yields

|S|v2(x) ≤∫S

v2dS+εδd−1

d

∫K

|x−y|1−dv2(y)dy+δd+1

εd

∫K

|x−y|1−d|∇v(y)|2dy.

Now integrating both sides with respect to x in K gives

|S|∫K

v2dK ≤ |K|∫S

v2dS +εδd

d

∫K

v2dK +δd+2

εd

∫K

|∇v|2dK,

which yields the desired estimate (A.26) by setting ε = |S|d2δd

. �

Consider the case of Lemma A.4 in which the convex domain K is a shape regulard-simplex. Denote by hK the diameter of K. The shape regularity implies that

(1) the measure of K is proportional to hdK ,

(2) there exists an inscribed ball BK ⊂ K with diameter proportional to hK .

Now let S be a ball inside of K with radius rS ≥ ς∗hK . Then, there exists a fixedconstant κ∗ such that

(A.31) |K| ≤ κ∗|S|.Apply (A.31) in (A.26) and notice that τ∗|S| ≥ hd

K and δ = hK . Thus,

(A.32) ‖v‖2K ≤ 2κ∗‖v‖2S +4τ2∗h

2K

d2‖∇v‖2K .

For simplicity of notation, we shall rewrite (A.32) in the following form:

(A.33) ‖v‖2K ≤ a0‖v‖2S + a1h2K‖∇v‖2K .

If v is infinitely smooth, then a recursive use of the estimate (A.33) yields thefollowing result:

(A.34) ‖v‖2K ≤n∑

j=0

ajh2jK‖∇jv‖2S + an+1h

2n+2K ‖∇n+1v‖2K .




In particular, if v is a polynomial of degree n, then

(A.35) ‖v‖2K ≤n∑

j=0

ajh2jK‖∇jv‖2S .

The standard inverse inequality implies that

‖∇jv‖S � h−jK ‖v‖S .

Substituting the above into (A.35) gives

(A.36) ‖v‖2K � ‖v‖2S .The result is summarized as follows.

Lemma A.5 (Domain Inverse Inequality). Let K ⊂ Rd be a d-simplex which hasdiameter hK and is shape regular. Assume that S is a ball in K with diameterrS proportional to hK ; i.e., rS ≥ ς∗hK with a fixed ς∗ > 0. Then, there exists aconstant C = C(ς∗, n) such that

(A.37) ‖v‖2K ≤ C(ς∗, n)‖v‖2Sfor any polynomial v of degree no more than n.

The usual inverse inequality in finite element analysis also holds true for piecewisepolynomials defined on the finite element partition Th provided that it satisfies theassumptions A1-A4 of Section 3.

Lemma A.6 (Inverse Inequality). Let Th be a finite element partition of Ω consist-ing of polygons or polyhedra. Assume that Th satisfies all the assumptions A1-A4as specified in Section 3. Then, there exists a constant C = C(n) such that

(A.38) ‖∇ϕ‖T ≤ C(n)h−1T ‖ϕ‖T , ∀T ∈ Th

for any piecewise polynomial ϕ of degree n on Th.Proof. The proof is merely a combination of Lemma A.5 and the standard inverseinequality on d-simplices. To this end, for any T ∈ Th, let S(T ) be the circumscribedsimplex that is shape regular. It follows from the standard inverse inequality that

‖∇ϕ‖T ≤ ‖∇ϕ‖S(T ) ≤ Ch−1T ‖ϕ‖S(T ).

Then we use the estimate (A.37), with K = S(T ), to obtain

‖∇ϕ‖T ≤ Ch−1T ‖ϕ‖S ≤ Ch−1

T ‖ϕ‖T ,where S is a ball inside of T with a diameter proportional to hT . This completesthe proof of the lemma. �

References

[1] D. N. Arnold and F. Brezzi, Mixed and nonconforming finite element methods: implemen-tation, postprocessing and error estimates (English, with French summary), RAIRO Model.Math. Anal. Numer. 19 (1985), no. 1, 7–32. MR813687 (87g:65126)

[2] Douglas N. Arnold, Franco Brezzi, Bernardo Cockburn, and L. Donatella Marini, Unifiedanalysis of discontinuous Galerkin methods for elliptic problems, SIAM J. Numer. Anal. 39(2001/02), no. 5, 1749–1779, DOI 10.1137/S0036142901384162. MR1885715 (2002k:65183)

[3] Ivo Babuska, The finite element method with Lagrangian multipliers, Numer. Math. 20(1972/73), 179–192. MR0359352 (50 #11806)

[4] L. Beirao da Veiga, K. Lipnikov, and G. Manzini, Arbitrary-order nodal mimetic discretiza-tions of elliptic problems on polygonal meshes, SIAM J. Numer. Anal. 49 (2011), no. 5,1737–1760, DOI 10.1137/100807764. MR2837482 (2012i:65229)



http://www.ams.org/mathscinet-getitem?mr=813687









[5] L. Beirao da Veiga, K. Lipnikov, and G. Manzini, Convergence analysis of the high-order mimetic finite difference method, Numer. Math. 113 (2009), no. 3, 325–356, DOI10.1007/s00211-009-0234-6. MR2534128 (2010g:65180)

[6] M. Berndt, K. Lipnikov, D. Moulton, and M. Shashkov, Convergence of mimetic finite dif-ference discretizations of the diffusion equation, East-West J. Numer. Math. 9 (2001), no. 4,265–284. MR1879474 (2003a:65085)

[7] Susanne C. Brenner and L. Ridgway Scott, The mathematical theory of finite element meth-

ods, Texts in Applied Mathematics, vol. 15, Springer-Verlag, New York, 1994. MR1278258(95f:65001)

[8] F. Brezzi, On the existence, uniqueness and approximation of saddle-point problems arisingfrom Lagrangian multipliers (English, with loose French summary), Rev. Francaise Automat.Informat. Recherche Operationnelle Ser. Rouge 8 (1974), no. R-2, 129–151. MR0365287(51 #1540)

[9] Franco Brezzi and Michel Fortin, Mixed and hybrid finite element methods, Springer Se-ries in Computational Mathematics, vol. 15, Springer-Verlag, New York, 1991. MR1115205(92d:65187)

[10] Franco Brezzi, Jim Douglas Jr., Ricardo Duran, and Michel Fortin, Mixed finite elements forsecond order elliptic problems in three variables, Numer. Math. 51 (1987), no. 2, 237–250,DOI 10.1007/BF01396752. MR890035 (88f:65190)

[11] Franco Brezzi, Jim Douglas Jr., and L. D. Marini, Two families of mixed finite ele-ments for second order elliptic problems, Numer. Math. 47 (1985), no. 2, 217–235, DOI10.1007/BF01389710. MR799685 (87g:65133)

[12] Franco Brezzi, Konstantin Lipnikov, and Mikhail Shashkov, Convergence of the mimeticfinite difference method for diffusion problems on polyhedral meshes, SIAM J. Numer. Anal.43 (2005), no. 5, 1872–1896 (electronic), DOI 10.1137/040613950. MR2192322 (2006j:65311)

[13] K. Chand and B. Henshaw, Overture Demo Introduction, 9th Overset Grid Symposium, PennState University.

[14] Philippe G. Ciarlet, The finite element method for elliptic problems, North-Holland Publish-ing Co., Amsterdam, 1978. Studies in Mathematics and its Applications, Vol. 4. MR0520174(58 #25001)

[15] Bernardo Cockburn, Jayadeep Gopalakrishnan, and Raytcho Lazarov, Unified hybridizationof discontinuous Galerkin, mixed, and continuous Galerkin methods for second order ellip-tic problems, SIAM J. Numer. Anal. 47 (2009), no. 2, 1319–1365, DOI 10.1137/070706616.MR2485455 (2010b:65251)

[16] Daniele Antonio Di Pietro and Alexandre Ern, Mathematical aspects of discontinuousGalerkin methods, Mathematiques & Applications (Berlin) [Mathematics & Applications],vol. 69, Springer, Heidelberg, 2012. MR2882148

[17] Jerome Droniou and Robert Eymard, A mixed finite volume scheme for anisotropic diffusionproblems on any grid, Numer. Math. 105 (2006), no. 1, 35–71, DOI 10.1007/s00211-006-0034-1. MR2257385 (2008d:65121)

[18] Paola Causin and Riccardo Sacco, A discontinuous Petrov-Galerkin method with Lagrangianmultipliers for second order elliptic problems, SIAM J. Numer. Anal. 43 (2005), no. 1, 280–302 (electronic), DOI 10.1137/S0036142903427871. MR2177145 (2006h:65185)

[19] Jaroslav Fort, Jirı Furst, Jan Halama, Raphaele Herbin, and Florence Hubert (eds.),Finite volumes for complex applications. VI. Problems & perspectives. Volume 1, 2, SpringerProceedings in Mathematics, vol. 4, Springer, Heidelberg, 2011. MR2882736 (2012h:00040)

[20] Raphaele Herbin and Florence Hubert, Benchmark on discretization schemes for anisotropicdiffusion problems on general grids, Finite volumes for complex applications V, ISTE, Lon-don, 2008, pp. 659–692. MR2451465 (2010a:65211)

[21] A. Khamayseh1 and V. Almeida, Adaptive hybrid mesh refinement for multiphysics applica-tions, Journal of Physics: Conference Series 78 (2007) 012039.

[22] L. Mu, J. Wang, and X. Ye, A weak Galerkin finite element methods with polynomial reduc-

tion, arXiv:1304.6481.[23] L. Mu, J. Wang, and X. Ye, Weak Galerkin finite element methods on polytopal meshes,

arXiv:1204.3655v2, submitted to International J. of Numerical Analysis and Modeling.[24] L. Mu, J. Wang, Y. Wang, and X. Ye, A computational study of the weak Galerkin

method for second order elliptic equations, Numerical Algorithms 63 (2013), 753–777.DOI:10.1007/s11075-012-9651-1.

































[25] L. Mu, J. Wang, X. Ye, and S. Zhao, A numerical study on the weak Galerkin method forthe Helmholtz equation with large wave numbers, Communication in Computational Physics,accepted.

[26] L. Mu, J. Wang, G. Wei, X. Ye, and S. Zhao, Weak Galerkin methods for second or-der elliptic interface problems, Journal of Computational Physics 250 (2013), 106–125.doi:10.1016/j.jcp.2013.04.042.

[27] P. Raviart and J. Thomas, A mixed finite element method for second order elliptic problems,

Mathematical Aspects of the Finite Element Method, I. Galligani, E. Magenes, eds., LecturesNotes in Math. 606, Springer-Verlag, New York, 1977.

[28] J. Wang, Mixed finite element methods, Numerical Methods in Scientific and EngineeringComputing, Eds: W. Cai, Z. Shi, C-W. Shu, and J. Xu, Academic Press.

[29] Junping Wang and Xiu Ye, A weak Galerkin finite element method for second order ellipticproblems, J. Comput. Appl. Math. 241 (2013), 103–115, DOI 10.1016/j.cam.2012.10.003.MR2994424

[30] Junping Wang and Xiu Ye, A weak Galerkin finite element method for the Stokes equation,arXiv:1302.2707, 2013.

Division of Mathematical Sciences, National Science Foundation, Arlington, Vir-

ginia 22230

E-mail address: [email protected]

Department of Mathematics, University of Arkansas at Little Rock, Little Rock,

Arkansas 72204

E-mail address: [email protected]


